A slightly messy mesh in blender 2.81 beta
In second case, try: Edit mode, vertex, A, alt+m -> Merge by distace
Merge should work. Take a closer look for double vertex.
In first case:
Option one, If the vertex was displaced only by z coordinate, I would just simply copy/pasted it from correct vertex. Or use this tutorial: https://www.youtube.com/watch?v=TnTFl51GGdw
Final thought: you can just select the mirror vertex and copy it's coordinates to the vertex you need to correct.
So I would just select the vertex in the correct spot on other side of the handle/x-coordinate, press N to bring out the side panel, go to item, copy coordinates and paste them into coordinates of corrupted vertex one by one, BUT changing the x coordinate from positive to negative/from negative to positive.